Morse Theory for Cell Complexes

Two-dimensional gauge theories revisited

This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex and its geometric dual. The derivation of these may require that the objects on the discrete mesh, but not the mesh itself, are interpolated. 
Our theory includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects. Definitions are given for discrete versions of all the usual operators of exterior calculus. The presence of forms and vector fields allows us to address their various interactions, which are important in applications. In many examples we find that the formulas derived from DEC are identical to the existing formulas in the literature. We also show that the circumcentric dual of a simplicial complex plays a useful role in the metric dependent part of this theory. The appearance of dual complexes leads to a proliferation of the operators in the discrete theory. 
One potential application of DEC is to variational problems which come equipped with a rich exterior calculus structure. On the discrete level, such structures will be enhanced by the availability of DEC. One of the objectives of this thesis is to fill this gap. There are many constraints in numerical algorithms that naturally involve differential forms. Preserving such features directly on the discrete level is another goal, overlapping with our goals for variational problems. 
In this thesis we have tried to push a purely discrete point of view as far as possible. We argue that this can only be pushed so far, and that interpolation is a useful device. For example, we found that interpolation of functions and vector fields is a very convenient. In future work we intend to continue this interpolation point of view, extending it to higher degree forms, especially in the context of the sharp, Lie derivative and interior product operators. Some preliminary ideas on this point of view are presented in the thesis. We also present some preliminary calculations of formulas on regular nonsimplicial complexes.

/pdf/discrete-exterior-calculus-1j71w2wt1y.pdf

Discrete exterior calculus

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

/pdf/a-roadmap-for-the-computation-of-persistent-homology-4gjwkkj6rv.pdf

A roadmap for the computation of persistent homology

A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us in this study. On the other hand, some very general theories have been developed for the study of smooth manifolds. One of the most powerful, and useful, of these theories is Morse Theory. In this paper we present a combinatorial adaptation of Morse Theory, which we call discrete Morse Theory, that may be applied to any simplicial complex (or more general cell complex). The goal of this paper is to present an overview of the subject of discrete Morse Theory that is sucient both to understand the major applications of the theory to combinatorics, and to apply the the theory to new problems. We will not be presenting theorems in their most recent or most general form, and simple examples will often take the place of proofs. We hope to convey the fact that the theory is really very simple, and there is not much that one needs to know before one can become a "user".

/pdf/a-user-s-guide-to-discrete-morse-theory-1v0wdvpbju.pdf

A user's guide to discrete morse theory

In this paper we study the zeta-function determinant in the context of elliptic boundary value problems. Our main technique is to relate the determinant of an operator, or a ratio of determinants, to the boundary values of the solutions of the operator. This has the advantage of restricting attention to the solutions of the operator, which do not depend on the boundary conditions and can often be written down explicitly, rather than the eigenvalues, which are usually difficult to work with. In addition, the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold. This has special significance in the case that the manifold is a finite interval. In this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix.

/pdf/functional-determinants-and-geometry-36sxvcq6xd.pdf

Functional determinants and geometry

In this paper we introduce the notion of a combinatorial dynamical system on any CW complex. Earlier in [Fo3] and [Fo4], we presented the idea of a combinatorial vector field (see also [Fo1] for the one-dimensional case), and studied the corresponding Morse Theory. Equivalently, we studied the homological properties of gradient vector fields (these terms were defined precisely in [Fo3], see also Sect. 2 of this paper). In this paper we broaden our investigation and consider general combinatorial vector fields. We first study the homological properties of such vector fields, generalizing the Morse Inequalities of [Fo3]. We then introduce various zeta functions which keep track of the closed orbits of the corresponding flow, and prove that these zeta functions, initially defined only on a half plane, can be analytically continued to meromorphic functions on the entire complex plane. Lastly, we review the notion of Reidemeister Torsion of a CW complex (introduced in [Re], [Fr]) and show that the torsion is equal to the value at 
$z=0$
 of one of the zeta functions introduced earlier. Much of this paper can be viewed as a combinatorial analogue of the work on smooth dynamical systems presented in [P-P], [Fra], [Fri1, 2] and elsewhere.

Robin Forman

Papers

Morse Theory for Cell Complexes

A user's guide to discrete morse theory

Functional determinants and geometry

Combinatorial vector fields and dynamical systems

Determinants of Laplacians on graphs